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The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.
We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillato
A three-dimensional harmonic oscillator with spin non-commutativity in the phase space is considered. The system has a regular symplectic structure and by using supersymmetric quantum mechanics techniques, the ground state is calculated exactly. We f
We compute the circuit complexity of scalar curvature perturbations on FLRW cosmological backgrounds with fixed equation of state $w$ using the language of squeezed vacuum states. Backgrounds that are accelerating and expanding, or decelerating and c
We study the bootstrap method in harmonic oscillators in one-dimensional quantum mechanics. We find that the problem reduces to the Diracs ladder operator problem and is exactly solvable. Thus, harmonic oscillators allow us to see how the bootstrap method works explicitly.
This is the contribution to Quarks2018 conference proceedings. This contribution is devoted to the holographic description of chaos and quantum complexity in the strongly interacting systems out of equilibrium. In the first part of the talk we presen